# The Algebra of Semi-flows: A Tale of Two Topologies

By K. Spendlove and R.C.A.M. Vandervorst

To capture the global structure of a dynamical system we reformulate dynam- ics in terms of appropriately constructed topologies, which we call *flow topologies*; we call this process *topologization*. This yields a description of a semi-flow in terms of a *bi-topological space*, with the first topology corresponding to the (phase) space and the second to the flow topology. A study of topology is facilitated through *discretization*, i.e. defining and examining appropriate finite sub-structures. Topol- ogizing the dynamics provides an elegant solution to their discretization by dis- cretizing the associated flow topologies. We introduce *Morse pre-orders*, an instance of a more general bi-topological discretization, which synthesize the space and flow topologies, and encode the directionality of dynamics. We describe how Morse pre-orders can be augmented with appropriate (co)homological informa- tion in order to describe invariance of the dynamics; this ensemble provides an *algebraization *of the semi-flow. An illustration of the main ingredients of the paper is provided by an application to the theory of discrete parabolic flows. Algebraiza- tion yields a new invariant for positive braids in terms of a bi-graded differential module which contains Morse theoretic information of parabolic flows. Finally, the main ingredients of this paper are illustrated in an example application to the theory of parabolic recurrence relations, and we show the algebraization of the parabolic semi-flow is a new invariant for positive braids.